SL 2.2—Functions and inverses

Treat a function as a rule with a defined input set.

A function assigns exactly one output to each allowed input; domain is inputs, range is resulting outputs, and the graph shows the relation.

Worked example
For f(x)=√x, the real domain is x≥0.

Worked example
Can one input have two outputs? not in a function.

Common boundary
A graph can fail the vertical-line test even if it looks like a curve.

Inverse example: if $f(x)=2x-3$, write $y=2x-3$, swap $x$ and $y$, and solve to get $f^{-1}(x)=(x+3)/2$. The graphs of $f$ and $f^{-1}$ reflect in $y=x$, and the domain of $f^{-1}$ is the range of $f$. An inverse function exists only after the original function is one-to-one on its stated domain.